The lyapunov exponent of vortex dynamics

Three-vortex system in an ideal fluid in the plane satisfies Hamiltonian form and the motion equations are integrable. However, the behavior of vertices are still complex so that it is difficult to study the passive particle in three point vertices system. Our focus is on the stability of passive particle with respect to an initial small perturbation, and Lyapunov exponent is introduced by employing Oseledec theory to describe the stability of passive particle quantitatively. In order to avoid fussy calculation, the simple expression of Lyapunov exponent is obtained by the conservation of volume in Hamiltonian system. Moreover, this definition leads to the partition of the instantaneous flow field in the point vortex system to show that the chaos motion of the passive particle only occurs in some especial regions.